I just figured the SortedLinkedList wouldn't have access to the Node's constructor.

The 'private' is to do with stuff accessing from outside of the outer class, not child classes, so in this case another class could not create a Node.
But Node is a part of a SortedLinkedList, as much as any member variable or method, so SortedLinkedList can execute private methods/constructors and access private attributes.
To be honest, since the Node class is 'private' itself, the declaration of the constructors as 'private' will have no practical effect. Nothing outside of SortedLinkedList could access it anyway.